Found problems: 85335
In a triangle $ABC$ with $\angle ACB = 90^{\circ}$ $D$ is the foot of the altitude of $C$. Let $E$ and $F$ be the reflections of $D$ with respect to $AC$ and $BC$. Let $O_1$ and $O_2$ be the circumcenters of $\triangle {ECB}$ and $\triangle {FCA}$. Show that: $$2O_1O_2=AB$$
Let \[ A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3) \].
Compute $2^A$.
Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?
Prove that all square matrices of the form (with $a, b \in R$),
$$\begin{pmatrix}
a & b \\
-b & a
\end{pmatrix}$$
form a commutative field $K$ when considering the operations of addition and matrix product. Prove also that if $A \in K$ is an element of said field, there exist two matrices of $K$ such that the square of each is equal to $A$.
The circles $ C_1,C_2 $ meet at the points $ A,B. $ A line thru $ A $ intersects $ C_1,C_2 $ at $ C,D, $ respectively. Point $ A $ is not on the arc $ BC $ of $ C_1, $ neither on the arc $ BD $ of $ C_2. $ On the segments $ CD,BC,BD $ there are the points $ M,N,K $ such that $ MN $ is parallel to $ BD $ and $ MK $ is parallel with $ BC. $ Upon the arc $ BC $ let $ E $ be a point having the property that $ EN $ is perpendicular to $ BC, $ and upon the arc $ BD $ let $ F $ be a point chosen so that $ FK $ is perpendicular to $ BD. $ Show that the angle $ \angle EMF $ is right.
A convex quadrilateral $ABCD$ Is placed on the Cartesian plane. Its vertices $A$ and $D$ belong to the negative branch of the graph of the hyperbola $y= 1/x$, the vertices $B$ and $C$ belong to the positive branch of the graph and point $B$ lies at the left of $C$, the segment $AC$ passes through the origin $(0,0)$. Prove that $\angle BAD = \angle BCD$.
(I, Voronovich)
Four distinct points are marked in a line. For each point, the sum of the distances from said point to the other three is calculated; getting in total 4 numbers.
Decide whether these 4 numbers can be, in some order:
a) $29,29,35,37$
b) $28,29,35,37$
c) $28,34,34,37$
One hundred friends, including Alice and Bob, live in several cities. Alice has determined the distance from her city to the city of each of the other 99 friends and totaled these 99 numbers. Alice’s total is 1000 km. Bob similarly totaled his distances to everyone else. What is the largest total that Bob could have obtained? (Consider the cities as points on the plane; if two people live in the same city, the distance between their cities is considered zero).
Suppose $\alpha,\beta,\gamma\in\{-2,3\}$ are chosen such that
\[M=\max_{x\in\mathbb{R}}\min_{y\in\mathbb{R}_{\ge0}}\alpha x+\beta y+\gamma xy\]
is finite and positive (note: $\mathbb{R}_{\ge0}$ is the set of nonnegative real numbers). What is the sum of the possible values of $M$?
$ \left(x_{1} x_{2} \ldots x_{1998} \right)$ shows a number with 1998 digits in decimal system. How many numbers $ \left(x_{1} x_{2} \ldots x_{1998} \right)$ are there such that $ \left(x_{1} x_{2} \ldots x_{1998} \right) \equal{} 7\cdot 10^{1996} \left(x_{1} \plus{} x_{2} \plus{} \ldots \plus{} x_{1998} \right)$ ?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$
Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have \[\frac{|R|}{|P|}\leq \sqrt 2\]
where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively.
[i]Proposed by Witold Szczechla, Poland[/i]
Determine all tuples of integers $(a,b,c)$ such that:
$$(a-b)^3(a+b)^2 = c^2 + 2(a-b) + 1$$
Let $A = \frac{1 \cdot 3 \cdot 5\cdot ... \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot ... \cdot (2n)}$ Prove that in the infinite sequence $A, 2A, 4A, 8A, ..., 2^k A, ….$ only integers will be observed, eventually.
Let $ABC$ be an acute triangle. Suppose the points $A',B',C'$ lie on its sides $BC,AC,AB$ respectively and the segments $AA',BB',CC'$ intersect in a common point $P$ inside the triangle. For each of those segments let us consider the circle such that the segment is its diameter, and the chord of this circle that contains the point $P$ and is perpendicular to this diameter. All three these chords occurred to have the same length. Prove that $P$ is the orthocenter of the triangle $ABC$.
(Grigory Galperin)
Prove that for all integers $ m$ and $ n$, the inequality
\[ \dfrac{\phi(\gcd(2^m \plus{} 1,2^n \plus{} 1))}{\gcd(\phi(2^m \plus{} 1),\phi(2^n \plus{} 1))} \ge \dfrac{2\gcd(m,n)}{2^{\gcd(m,n)}}\]
holds.
[i]Nanang Susyanto, Jogjakarta [/i]
Find all positive integer values of $n$ such that the value of the
$$\frac{2^{n!}-1}{2^n-1}$$
is a square of an integer.
Michael and James are playing a game where they alternate throwing darts at a simplified dartboard. Each dart throw is worth either 25 points or 50 points. They track the sequence of scores per throw (which is shared between them), and on the first time the three most recent scores sum to 125, the person who threw the last dart wins. On each throw, a given player has a $2/3$ chance of getting the score they aim for, and a $1/3$ chance of getting the other score. Suppose Michael goes first, and the first two throws are both 25. If both players use an optimal strategy, what is the probability Michael wins?
[i]Proposed by Michael Duncan[/i]
Let triangle $ABC$ have altitudes $BE$ and $CF$ which meet at $H$. The reflection of $A$ over $BC$ is $A'$. Let $(ABC)$ meet $(AA'E)$ at $P$ and $(AA'F)$ at $Q$. Let $BC$ meet $PQ$ at $R$. Prove that $EF \parallel HR$.
[i]Proposed by Daniel Hu[/i]
On the coordinate plane, the parabola $y = x^2$ and the points $A(x_1, x_1^2)$, $B(x_2, x_2^2)$ are set such that $x_1=-998$, $x_2 =1999$ The segments $BX_1$, $AX_2$, $BX_3$, $AX_4$,..., $BX_{1997}$, $AX_{1998}$ and $X_k$ are constructed succesively with $(x_k,0)$, $1 \le k \le 1998$ and $x_3$, $x_4$,..., $x_{1998}$ are abscissas of the points of intersection of the parabola with segments $BX_1$, $AX_2$, $BX_3$, $AX_4$,..., $BX_{1997}$, $AX_{1998}$. Find the value $\frac{1}{x_{1999}}+\frac{1}{x_{2000}}$
On Thursday 1st January 2015, Anna buys one book and one shelf. For the next two years she buys one book every day and one shelf on alternate Thursdays, so she next buys a shelf on 15th January. On how many days in the period Thursday 1st January 2015 until (and including) Saturday 31st December 2016 is it possible for Anna to put all her books on all her shelves, so that there is an equal number of books on each shelf?
Set $M=\{1,2,\cdots,1000\}$, for any $X\subseteq M(X\neq\varnothing)$, define $a_X$: sum of the minumum and maximum number in $X$. Then, the arithmetic mean of all $a_X$ is________.
Let $\Delta A_1B_1C$ be a triangle with $\angle A_1B_1C = 90^{\circ}$ and $\frac{CA_1}{CB_1} = \sqrt{5}+2$. For any $i \ge 2$, define $A_i$ to be the point on the line $A_1C$ such that $A_iB_{i-1} \perp A_1C$ and define $B_i$ to be the point on the line $B_1C$ such that $A_iB_i \perp B_1C$. Let $\Gamma_1$ be the incircle of $\Delta A_1B_1C$ and for $i \ge 2$, $\Gamma_i$ be the circle tangent to $\Gamma_{i-1}, A_1C, B_1C$ which is smaller than $\Gamma_{i-1}$.
How many integers $k$ are there such that the line $A_1B_{2016}$ intersects $\Gamma_{k}$?
Let $ABC$ be an acute triangle inscribed in a circle centered at $O$. Let $M$ be a point on the small arc $AB$ of the triangle's circumcircle. The perpendicular dropped from $M$ on the ray $OA$ intersects the sides $AB$ and $AC$ at the points $K$ and $L$, respectively. Similarly, the perpendicular dropped from $M$ on the ray $OB$ intersects the sides $AB$ and $BC$ at $N$ and $P$, respectively. Assume that $KL=MN$. Find the size of the angle $\angle{MLP}$ in terms of the angles of the triangle $ABC$.
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point $(2021,2021)$?
Determine the number of functions $f: \mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N} \to \mathbb{N}$ such that for all $n \in \mathbb{N}$:
\[
f(g(n)) = n + 2015,
\]
\[
g(f(n)) = n^2 + 2015.
\]